Negative moments as the signature of the radial density at small distances

TL;DR

This paper introduces a method using negative-order radial moments in momentum space to evaluate the behavior of radial densities at small distances, emphasizing the importance of form factor measurements at high momentum transfer.

Contribution

It presents a novel approach for assessing radial densities near zero distance, including regularization schemes and applications to nucleon form factors, both non-relativistic and relativistic.

Findings

Validated method against analytical Maclaurin expansion for non-relativistic cases.

Applied approach to relativistic Dirac form factor where expansion may not exist.

Highlighted the significance of high momentum transfer measurements for small-distance density evaluation.

Abstract

The present paper proposes a robust evaluation of any radial density at small distances using negative-order radial moments evaluated in momentum space. This evaluation provides a valuable insight into the behavior of a given radial density in the vicinity of $r=0$, and puts strong emphasis on the importance of measuring form factors at large squared four-momentum transfer, a domain essential for the determination of negative order moments. A specific attention is paid to the regularization scheme directly affecting the numerical determination of the radial density's parametrization. The proposed method is applied to non-relativistic study cases of the nucleon electric ($G_{En}, G_{Ep}$), and proton magnetic $G_{Mp}$ form factors. The validation is performed through comparison of the results of the approach to the analytically determined Maclaurin expansion - in the vicinity of $r=0$ - of the radial density function. The method is also applied to the relativistic Dirac form factor $F_1$ of the proton. In such a non-trivial case, the Maclaurin development might not exist for the radial density, rendering the determination from the proposed method extremely important.